Author Martin Fürer

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Martin Fürer

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Martin Fürer. Multi-Clique-Width. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


Multi-clique-width is obtained by a simple modification in the definition of clique-width. It has the advantage of providing a natural extension of tree-width. Unlike clique-width, it does not explode exponentially compared to tree-width. Efficient algorithms based on multi-clique-width are still possible for interesting tasks like computing the independent set polynomial or testing c-colorability. In particular, c-colorability can be tested in time linear in n and singly exponential in c and the width k of a given multi-k-expression. For these tasks, the running time as a function of the multi-clique-width is the same as the running time of the fastest known algorithm as a function of the clique-width. This results in an exponential speed-up for some graphs, if the corresponding graph generating expressions are given. The reason is that the multi-clique-width is never bigger, but is exponentially smaller than the clique-width for many graphs. This gap shows up when the tree-width is basically equal to the multi-clique width as well as when the tree-width is not bounded by any function of the clique-width.
  • clique-width
  • parameterized complexity
  • tree-width
  • independent set polynomial
  • graph coloring


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  1. Stefan Arnborg, D. G. Corneil, and Andrzej Proskurowski. Complexity of finding embeddings in a k-tree. SIAM Journal of Alg. and Discrete Methods, 8:277-284, 1987. Google Scholar
  2. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996. URL:
  3. Hans L. Bodlaender, Pål G. Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov, and Michal Pilipczuk. An O(c^k n) 5-approximation algorithm for treewidth. In Proc. 54th FOCS 2013, pages 499-508. IEEE, 2013. Google Scholar
  4. Hans L. Bodlaender, John R. Gilbert, Hjálmtyr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms, 18(2):238-255, 1995. URL:
  5. Binh-Minh Bui-Xuan, Jan Arne Telle, and Martin Vatshelle. Boolean-width of graphs. Theor. Comput. Sci., 412(39):5187-5204, 2011. URL:
  6. Derek G. Corneil and Udi Rotics. On the relationship between clique-width and treewidth. SIAM J. Comput., 34(4):825-847, 2005. URL:
  7. Bruno Courcelle. The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Inf. Comput, 85(1):12-75, March 1990. URL:
  8. Bruno Courcelle, Joost Engelfriet, and Grzegorz Rozenberg. Handle-rewriting hypergraph grammars. J. Comput. Syst. Sci., 46(2):218-270, 1993. URL:
  9. Bruno Courcelle and Johann A. Makowsky. Fusion in relational structures and the verification of monadic second-order properties. Mathematical Structures in Computer Science, 12(2):203-235, 2002. Google Scholar
  10. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. URL:
  11. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discrete Applied Mathematics, 108(1-2):23-52, 2001. URL:
  12. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discrete Applied Mathematics, 101(1-3):77-114, 2000. URL:
  13. Bruno Courcelle and Andrew Twigg. Constrained-path labellings on graphs of bounded clique-width. Theory Comput. Syst., 47(2):531-567, 2010. URL:
  14. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL:
  15. Michael R. Fellows, Frances A. Rosamond, Udi Rotics, and Stefan Szeider. Clique-width minimization is NP-hard. In Jon M. Kleinberg, editor, STOC, pages 354-362. ACM, 2006. URL:
  16. Eldar Fischer, Johann A. Makowsky, and Elena V. Ravve. Counting truth assignments of formulas of bounded tree-width or clique-width. Discrete Applied Mathematics, 156(4):511-529, 2008. URL:
  17. Fedor V. Fomin, Daniel Lokshtanov, Michal Pilipczuk, Saket Saurabh, and Marcin Wrochna. Fully polynomial-time parameterized computations for graphs and matrices of low treewidth. CoRR, abs/1511.01379, 2015. URL:
  18. Martin Fürer. A natural generalization of bounded tree-width and bounded clique-width. Proceedings of LATIN 2014: Theoretical Informatics - 11th Latin American Symposium. Springer LNCS, 8392:72-83, 2014. URL:
  19. Robert Ganian and Petr Hlinený. On parse trees and myhill-nerode-type tools for handling graphs of bounded rank-width. Discrete Applied Mathematics, 158(7):851-867, 2010. URL:
  20. Petr Hlinený and Sang il Oum. Finding branch-decompositions and rank-decompositions. SIAM J. Comput, 38(3):1012-1032, 2008. URL:
  21. Sang-il Oum. Approximating rank-width and clique-width quickly. ACM Trans. Algorithms, 5(1):10:1-10:20, December 2008. URL:
  22. Sang-il Oum and Paul D. Seymour. Approximating clique-width and branch-width. J. Comb. Theory, Ser. B, 96(4):514-528, 2006. URL:
  23. G. Pólya and R. C. Read. Combinatorial Enumeration of Groups, Graphs and Chemical Compounds. Springer-Verlag, New York, 1987. Google Scholar
  24. Neil Robertson and Paul D. Seymour. Graph minors. III. Planar tree-width. J. Comb. Theory, Ser. B, 36(1):49-64, 1984. URL:
  25. Sigve Hortemo Sæther and Jan Arne Telle. Between treewidth and clique-width. In Dieter Kratsch and Ioan Todinca, editors, Graph-Theoretic Concepts in Computer Science - 40th International Workshop, WG 2014, Nouan-le-Fuzelier, France, June 25-27, 2014. Revised Selected Papers, volume 8747 of Lecture Notes in Computer Science, pages 396-407. Springer, 2014. URL: